Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [inverse-function]

For questions regarding an inverse function as the dominant topic of the post, or for questions requesting guidance on finding the inverse function for a particular function.

In mathematics, an inverse function or $f^{-1}$ is a function that "reverses" another function. That is, if $f$ is a function mapping $x$ to $y$, then the inverse function of $f$ maps $y$ back to $x$.

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How do we know that is not possible to invert $x=t+\cos t$ analytically?

From this answer to how to get a get a nice “cosine looking” curve following the y=x direction? It is not possible to invert $x=t+\cos t$ analytically. I'm sure it's true and I wouldn't know how to try, but how can the impossibility be shown?
uhoh
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Can a non-surjective function have an inverse?

I am confused by the many conflicting answers/opinions at e.g. 1, 2. Many claim that only bijective functions have inverses (while a few disagree). So, for example, does $f:\{0\}\rightarrow \{1,2\}$ defined by $f(0)=1$ have an inverse? According to…
user629687
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why $\sin^{-1}(x)+\cos^{-1}(x) = π/2$

How do i find the When $x$ is ranging from $-1$ to $1$? I want know why $\sin^{-1}(x)+\cos^{-1}(x) = π/2$ I have already tried inverse-function. thanks.
장원봉
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Calculating inverse trigonometric values without a calculator (AEA 2016)

Find the value of $$\arccos(1/\sqrt2) + \arcsin (1/3) + 2 \arctan(1/\sqrt2).$$ Give your answer as a multiple of $\pi$. This was the least well answered question on Edexcel's Advanced Extension Award annual paper in 2016. The next paper is…
user634745
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Looking for a simple interpretation

$f(x) = 10-x$ If I plugin $x=2$, I get $f(2)=10-2=8$. If I want to know what I must plugin to get $8$, again I simply plugin $8$ into $f(x)$ : $f(8) = 10-8=2$. One can conclude $f(x) = f^{-1}(x)$. But this is not so clear to me. Specifically for…
AgentS
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Can the inverse of this function be expressed in closed form?

So there is such a visual way to show that set of integers has the same cardinality as the set of natural numbers. $$0,1,-1,2,-2,3,-3,4,-4,5,-5...$$ But I think it is not really rigorous proof (and I think it does not pretend to be so) of the fact…
Юрій Ярош
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Inverse of Strictly Monotone Function

Let $f:\mathbb{R}\to\mathrm{S}$, where $\mathrm{S}\subseteq\mathbb{R}$, be a strictly increasing (decreasing) function. I wish to find out whether its inverse $f^{-1}$ is also strictly increasing (decreasing) or not. While a proof for this exists…
index
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Mirror function with respect to a straight line

I have a question for which I have unfortunately not come up with an answer so far. If I want to mirror a function on the bisector, then $f(x)=y$ applies and then I have the inverse. So far so good. But what do I do now if I want to mirror a…
manofthousandnames
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What is the inverse for: $x^2+4x+5,x\ge-2$

What is the inverse for: $x^2+4x+5,x\ge-2$ I have tried to solve it using the quadratic formula: $x^2+4x+5=y$ $x^2+4x+5-y=0$ $x=-2\pm\sqrt{y-1}$ $y\ge1$ The inverse should be: $x=-2+\sqrt{y-1}$ I don't understand why it sould be $+\sqrt{y-1}$ and…
Ridertvis
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Inverting a cubed root equation with two cube roots

In a recent StackOverflow question, I came across this formula: $$up = (1 + p)^\frac{2}3 - (1 - p)^\frac{2}3 $$ and the question required the poster to plot p as a function of u. I misunderstood the question and thought it was just a simple…
ewokx
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Evaluate $\tan(\frac{1}{2}\sin^{-1}\frac{3}{4})$

Evaluate $\tan(\frac{1}{2}\sin^{-1}\frac{3}{4})$ My attempt: Method:- 1 Let $\sin^{-1}\frac{3}{4} = \theta \implies \sin\theta = \frac{3}{4}$ and $\theta \in [0,\frac{π}{2}]$ Therefore $$\cos\theta = \sqrt{1-\sin^2\theta} = \frac{\sqrt 7}{4}$$ Now…
Largest Prime
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What does $\{g:\mathbb{R}\rightarrow\mathbb{R}\mid g\circ f=f\circ g\}$ mean?

Is the set $\{g:\mathbb{R}\rightarrow\mathbb{R}\mid g\circ f=f\circ g\}$ infinite? Why? Let's suppose $f(x)=x^2$ $g(x)$ is an inverse function of $f(x)$. $g(f(x))=(x^2)^{\frac12}=x$ Therefore, since composition of functions are…
Nay Sie
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Inverting a non-analytic exponential

I've been recently interested in the nth derivative of $x^x$, and I jumped to pondering of the inverse of not this function, (because it's known to be $e^{\operatorname{W}\left(\ln x\right)}$, for $\operatorname{W}\left(x\right)$ being the Lambert W…
Vessel
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By restricting the domain of a non-bijective function, can we still find the inverse function?

Suppose we have a cubic polynomial function $f:\mathbb{R}\to\mathbb{R}$ given by $$y=f(x)=ax^3+bx^2+cx+d,$$ where $a,b,c,d\in\mathbb{R}$ and $a\neq 0$. As far as I know there are two ways to find the inverse function of $f$, namely the Cardano…
johnny09
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ways to invert a specific function

Given function $y = e^x$, the inverse is $x = \ln{y}$. Is it possible to find the inverse of $y = x*e^x$? If this is not possible mathematically, are tools like programs or numerical methods available to do it? I want to put the result, that is, x…
umdas
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